Circuits embodying negative impedance



Nov. 23, 1.937. H. NYQUIST 2,099,759

CIRCUITS EMBODYING NEGATIVE IMPEDANCE Filed June 23, 1936 2 Sheets-Sheet- 1 FIG! 5 F/GZ FIG 5 F G 6 wag/$1 4k) MAG/#41?! C ma 553% I Has IHA GINARY A 7'7'ORNEV INVENTOR By HNVQU/ST Patented Nov. 23, 1937 UNITED STATES PATENT OFFICE CIRCUITS EIHBODYING NEGATIVE IMPEDANCE poration of New York Application June 23, 1936, Serial No. 86,882

4 Claims.

The present invention relates to circuits which consist in part of negative impedance elements.

It is an object to so design and proportion the negative impedance elements that the network as a whole shall be stable. It is a further object of the invention to adjoin to a given positive impedance network having undesirable impedance characteristics such a negative impedance as will neutralize all or a part of the undesirable characteristics.

It is well known by those familiar with the properties of negative impedances that the combination of negative impedance elements and positive impedance elements in a common network may result in a condition of instability. It is also known that negative impedances may be classified into two types, the series type and the shunt type. It is further well understood that when a positive resistance element and a negative resistance are connected at both terminals the conditions for stability are that the positive impedance should be numerically greater than the negative impedance if the latter is of the series type and should be numerically smaller than the negative impedance if the latter is of the shunt type. For a fuller statement of these facts-the reader may be referred to an article Negative Impedance and the Twin Zl-Type Repeater by George Crisson published in the Bell System Technical Journal, volume 10, page 485, July 1931.

In addition to the negative resistance elements there may be reactive negative impedance elements and negative elements which are partly resistive and partly reactive. Moreover, the positive networks to which they may be connected may be partly or wholly reactive. In order to make use of negative impedance elements in combination with positive elements it is therefore necessary to know how to proportion the negative elements for more general combinations than those hitherto studied and to know whether the series or shunt type should be used in any specific case.

In order to make the instructions for design intelligible a brief theoretical discussion of nega tive impedances will first be given. This discussion will then be set forth in rules which will enable those skilled in the art of circuit design to include negative impedance elements Without incurring instability.

Reference is now made to the accompanying drawings in which:

Figure 1 is a schematic representation of a negative impedance of the series type.

Figure 2 is a schematic representation of a negative impedance of the shunt type.

Fig. 3 represents the series connection of two impedances and a source of electromotive force.

Fig. 4 represents the series connection of a positive inductance, a positive resistance and a negative resistance.

Figs. 5, 6, '7, 8 and 9 represent mathematical curves to be fully explained hereinafter.

Fig. 10 represents an electrical network having one negative impedance member and.

Figs. 11, 12 and 13 illustrate the use of negative impedances in connection with a transmission line.

A negative impedance is characterized by having an internal source of energy. This source of energy may be of difierent form in difierent impedance elements but it must be such that it is controlled from the terminals. This control may be of either of two kinds. Either the impressed current may control the internal source of energy so as to produce an electromotive force, in which case the negative impedance is said to be of the series type, or the impressed electromotive force may control the internal source of energy to produce a current at the terminals in which case the negative impedance is said to be of the shunt type.

Referring now more particularly to Fig. 1, which represents schematically a negative impedance of the series type I is a current actuated device, and 2 is an internal electromotive force whose value is controlled by the current actuated device, the control being indicated by the arrow 3. The elements I and 2 are to be considered free of internal impedance and any actual impedance which they have may be considered lumped in series with both as at 4. The figure may be made clearer by considering a specific approximation to a negative impedance namely a series wound dynamo generator running at a uniform speed. Then I is the field winding and 2 is the armature and commutator. The arrow 3 indicates the well known property of such machines whereby the current in the winding determines the electromotive force appearing at the commutator terminals. Finally, 4 is the sum total of the internal impedance of the machine, which is positive but normally numerically small in comparison with the negative impedance of the machine.

In a similar manner Fig. 2 represents schematically a negative impedance of the shunt type. The element is an electromotive force actuated device and 8 is a source of internal current whose magnitude is determined by the electromotive force across i. The control is indicated by the arrow 9. Both 1 and 8 are to be considered free of internal admittance and any internal admittance which'they may have may be considered lumped as at Hi.

If the mechanism of the impedance is understood then a comparison with Figs. 1 and 2 will establish whether any given negative impedance is of the series or the shunt type. It may, however, be necessary to deal with negative impedance whose mode of operation is not well understood and a more general criterion for type is necessary. An inspection of Figs. 1 and 2 will show that the series type impedance is stable when open circuited (or connected to a sufiiciently great positive impedance) and is unstable when short-circuited (or connected to a sufficiently small positive impedance), and that the shunt type impedance is stable when short-circuited and unstable when open circuited. These properties may then be used'as criteria of types. I In any known system the control indicated by 3 in Fig. 1 or by 9 in Fig. 2 is not strictly instane taneous.. As the frequency increases indefinitely the control utilmately dwindles and disappears. In Fig. 1 this means that the ratio of generated electromotive force to controlling current approaches zero. In Fig. 2 it means that the ratio of. generated current to controlling electromotive force approaches zero. This property may be expressed by saying that a negative impedance of the series type contains a factor 1 which is in the nature of a positive transfer admittance, which is substantially unity at frequencies of practical interest and which approaches zero as the frequency increases indefinitely, and that a negative impedance of the shunt type contains a factorl/ where 1; is defined as before.

We are now in a position to define negative impedanc'es of both types quantitatively. An impedance 21 will be said to be negative and of the series type if 21= Z1; and it is stable when open circuited and unstable when short circuited; and an impedance 22 will be said to be negative and of the shunt type if it satisfies the condition 22=Z/'r and is stable when short circuited and unstable when open circuited. Z is a positive impedance and v is a factor of the form discussed above. In what follows 7 will be limited to the form where e is the base of natural logarithm,

and e is a positive number, which may be taken arbitrarily small. This restriction is sufficient for practical purposes. It is not necessary here to prove that negative impedances are necessarily such as to be unstable either when open-circuited or when short circuited. It is however, desirable to point out that all impedances which approach to the property of being negative, are of one or the other type as far as existing literature discloses. I

Referring now more particularly to Fig. 3, the diagram represents a network of impedances Z .and z connected in series with the electromotive force 605). (t) varies with time so as to ap proach zero as 15 increases indefinitely. In order to investigate the stability of the network it is necessary and sufficient to determine whether under these conditions the current i(t) arising in the network as a response to 203) also approaches zero as it increases indefinitely. The case will first be considered when Z is a positive impedance and z is a negative impedance of the series type. Since 2 is stable when open circuited and unstable when short circuited it follows that there is no meaning in inquiring what current will fiow in z in response to a given electromotive force but there is meaning in inquiring what electromotive force E will appear across'the terminals as a result of given current. The necessary analysis will be carried out as a series of successive approximations which avoids the former inquiry. It will be assumed that the reader is familiar with my article Regeneration Theory published in Bell System Technical Journal. volume 11, page 126, January 1932. The notation below will be the same as the one there used except that [L is used instead of z.

' Let E be defined by sn f -Jan en 1) As a first approximation we will compute the current that would flow if 2 were made equal to zero. It is This procedure for computing the current for a given voltage is justified, for example, in the article The Practical Application of the Fourier Integral by G. A. Campbell in B. S. T. J. for October 1928. The eifect E(g) is my i003); the cause C(g) is my e(t). The function Y is to be taken as 1/2 since the cause is a voltage and the effect is a current. Particular reference is made to the equations near the top of page 656, together with pairs 101 and 102 on page 663. Reference is also made to applicants article referred to, particularly Equation (11) for the relationship between the integral here used and the one treated by Campbell. Next we may assume this current flowing in the circuit and inquire what electromotive force it produces in z. This electromotive force can be applied to Z and a correction term for the current can thus be computed. This can be computed by the same processes as were used in Equation (2) with the distinction that the current is now to be take-n as the cause and the voltage as the effect, and therefore the appropriate function to use for Y is 2. We have for the drop resulting from io(t) V and for the resulting correction term to the current 1 1 (t)= ;;.J Eze d /Z Similarly this correction term gives rise to an electromotive force across a and this gives rise to the current The counterelectromotive force resulting when current flows through an impedance is of opposite sign to the electromotive force which causes that current to flow. Hence in adding these current components alternate components have their sign reversed. The total current up to and including the nth correction term is:

The presence of the factor hfi e in z insures that the last of these integrals approaches zero as 11 increases. We have then for the total current the value which the series approaches as n is increased indefinitely, namely:

' tive impedance of the series type their combination is stable or unstable according to whether or not the point (1,0) lies outside the locus of (z/Z).

By the locus of any function f(w) is here to be understood the curve obtained by plotting plus and minus the imaginary part of f(w) as a function of the real part for all real and positive values of w.

By a similar process I have found that when 2 is negative and of the shunt type the following rule holds good:

II. If Z is a positive impedance and z is a negativeimpedance of the shunt type, then their combination is stable or unstable according to whether or not the point (1,0) lies outside the locus of (Z/z).

I have also found by comparing these two rules that the following rule is true:

III. If a positive impedance Z and a negative impedance 2 form a stable combination then the change of type of 2 results in an unstable combination.

This rule emphasizes the necessity of specifying the type of impedance as well as its numerical value. If the type is left unspecified one at least of the two possibilities will be certain to be unstable.

It is sometimes desirable to make up networks containing a plurality of positive or negative impedances. This case is not covered by the rules just given. By applying the methods given above to this more general case I have found a rule for this case also which will now-be stated.

Let the general network consist of h impedance branches and Q1 independent junction points and let the positive part of the net work.

Write g-l equations of the form pq q (Iva) expressing the fact that the total current flowing The stability rule then is:

IV. If the origin lies outside of the locus of D, the circuit is stable. If not, it is unstable.

It sometimes happens when some of the impedance elements are pure reactances that the quantity to be plotted, 2/Z, Z/g, or D takes on infinite values and thus cannot be plotted. I have found that the plots can be completed and that they lead to correct results if a small resistance or a small conductance is included with the reactance. For example, a capacitance has an infinite impedance (1 at zero frequency and therefore an expression which has this impedance in the numerator cannot be plotted. If an arbitrarily small conductance is bridged around the condenser, the result can be plotted. Similarly, a pure inductance has zero impedance at zero frequency and an expression having it as a factor in the denominator cannot be plotted. By adding a small series resistance this difllculty can be overcome.

The application of the rules enumerated can be made clearer by few specific examples. Referring to Fig. l, I3 is a coil Whose impedance is sufficiently well represented by an inductance L and a resistance R, both positive, connected in series. Now, suppose it has been decided that R has been found greater than desirable and it has been proposed to reduce it by connecting the negative resistance, M, in series therewith. By negative resistance is to be understood an impedance having the value m or 1/ as the case may be, 1 being real and positive and n being as described hereinbefore.

Referring to Fig. 5 curve A represents the locus for the case where the negative resistance is of the series type and r is numerically greater than R. The point (1,0) lies inside of this curve and therefore the combination is unstable (Rule I, above). Curve B represents the curve for the case when the negative resistance is of the series type and R is numerically greater than 7'. The point (1,0) lies outside of this curve and the combination is stable (Rule I).

Fig. 6 represents the locus when the negative impedance is of the shunt type (Rule II). Here the point (1,0) is seen to lie inside of the locus regardless of whether 2' is great or small in comparison with R. With a negative impedance of the shunt type, then, the proposed circuit is always unstable. I have thus found that a negative resistance for the purpose contemplated must be numerically smaller than the positive resistance, which might have been suspected from the known facts when a negative and a positive resistance both without reactance are connected together. I have furthermore found, however, that the negative impedance must not be of the shunt type, no matter what its value, which would not be suspected from the known factor about connecting resistances together.

The question might be under consideration of connecting negative capacitance across a telephone line to neutralize some or all of the natural line capacitance. In order to exemplify the methods herein described this case may now be considered. As a first step, let the line he distortionless; that is, let its lineconstants, L,

C, R and G be uniformly distributed and so related that L/C /RG. It is well known that the characteristic impedance being K, and the line east and the line west being effectively in parallel, and 2=[-/iwC']'n. The quantity to be plotted is then Z'q/iwCK. This expression is infinite for 0:0, so that according to the instructions' given above we may add the extremely small conductance -g across the capacitance making the quantity to be plotted (21 /K(iwC+g) Referring now to Fig. 7, the plot for this quantity shows that (1,0) lies inside and the circuit is unstable for all values of C. Therefore, no negative capacitance of the series type may be connected across a positive resistance.

If next the negative capacitance is of the shunt type, Rule II applies and we have Z=1/2'@C1 and the quantity to be plotted is Z/2:iwCK1 /2. This plot is shown in Fig. 8 and again shows the point (1,0) on the inside for any value of C. Hence, no negative capacitance of the shunt type may be connected across a positive resistance.

The proposal to connect a negative capacitance across a line which isalready distortionless has then been found fruitless because both types of negative capacitance lead to an unstable condition regardless of how small or how large the value. case where the line is not distortionless to begin with but has an excess of capacity. To illustrate the principles, it is sufficient to answer to the question: Under what conditions may a negative capacitance be bridged across a positive impedance consisting of a resistance and a capacitance in parallel? The consideration of the case in which the capacitance is of the series type is not essentially different from the considerations which led to Fig. '7, and the conclusion is the same as there arrived at, namely, that no negative capacitance of the series type is permissible. If the negative capacitance is of the shunt type, we'have for the application of Rule II that Z=1(iwc+g), z= M1001; and the quantity to be plotted is Z/z=iwC'1;/(iw0+g). The resultant plotsare shown in Fig. 9 for two conditions. Curve A represents the case in which C/c 1, and shows an unstable condition. Curve B represents the case in which O/c 1, and shows a stable condition. Hence, a negative capacitance of the shunt type may beused, provided it is numerically smaller than the positive capacitance. To sum up: Across a positive network consisting of a resistance and a capacitance in parallel,

there can be connected no negative capacitance The next step to consider then is the.

' stable.

of the series type, but a capacitance of the shunt type can be connected, provided it is numerically less than the positive capacity. 7

Similarly, it may be proposed to bridge a negative capacitance across a positive network conany proposed network'and determine whether or not it is stable. It should be specifically pointed out that Rule IV permits the inclusion of a plurality of negative. impedances of either or both types. a

Most of the negative impedances described in the literature on the subject are such that the designer can determine their type by a comparison with Figs. 1 and 2, described above. There are cases, however, where the negative impedances are constructed out of a negative resistance and positive impedance elements. A discussion of such a case will be given with the two-fold object of enabling the designer to determine the type in similar cases and of giving one more illustration of the stability rules.

Referring now to Fig. 10, numerals l5 and l6 represent the terminals of a network made up of a positive impedance W, two positive resistancesr and a negative resistance 'r (strictly r1 'or r/1 It can be shown by straightforward calculations that the impedance-of such a network is' -r/ W. It is therefore a negative impedance. To determine its type assume first that the resistance -r is of the series type. With the terminals l5, [6 short circuited the network consists of (r), r, and W in parallel. Applying Rule I it is apparent that z/Z is greater'than +1 for w=0 and Z/Z=0 fOr'w=.

Works will be readily able to construct a locus'for The point (1,0) is therefore inside the locus and there is instability. Similarly when the terminals are open circuited z/Z is not positive and greater than unity for any value of w. The circuit is therefore The whole network being stable when open circuited and unstable when short-circuited is of the seriestype. Similarly when the resistance element '(1) is of the shunt type the whole network is of the shunt type.

An example will now be given of an application of Rule IV. It will be understood that the application of this rule will generally result in rather lengthy calculations. In order to reduce the discussion the case taken is simplified by being limited to two negative impedances and in other respects. It does not however omit anything that is essential in the study of more complex cases. I

In Fig. 11 let I and 2 be the conductors of a uniform line and let 23 and 24 be impedances terminating it. Let a and b be sections of the line situated at distances equal to one-fourth of the entire distance from the ends and let-it be;

line is such that RC LG and that the negative i impedances should be of'the shunt type.

As a first step find .a 11 network which is equivalent to the portion of the line between a and b and find impedances equivalent to the portion to the left of a and similarly to the right of b including the terminating impedances. The methodvof making such computations are given for example in the book Transmission Circuits for Telephone Communication by K; S. Johnson. After these computations. have been made the problem under consideration is 'equivalent to the problem of the network shown in Fig. 12, where 211 and 231 are equivalent to the portions of the given circuit to the left of a a'ndto the right of b respectively, 212, 22, 232 are equivalent to the portion between a and b, and 24 and 25 are the negative impedances of the problem. Next compute the impeda'nces 21 and 23 which are equivalent to 211 and 212 and 231 and 232 in parallel respectively. The new equivalent network is shown in Fig. 13. V 7

At the junction of 21 24 and .22 the following equation expresses the fact that the net current converging there is zero Similarly for the junction of 22, 25, and 23 The following three equations express the fact that the net electromotive force around the three meshes shown is zero in each case.

0 -1 1 0 1 Z1 0 0 z4 0 0 22 0 Z4 Z5 0 0 23 0 Z5 which in accordance with a well known rule of algebra may be written:

In order to obtain the quantity D to be plotted this expression is to be divided by 2425 since both the negative impedances are to be of the shunt type.

Also on account of the assumed symmetry we.

have 21:23 and 24:25 we then have If neither factor when plotted results in a curve which encloses the origin it is clear that the product also does not result in a curve which encloses the origin. Conversely it can'be shown that'if either factor (or both) givesa curve which encloses the origin so does the product. Now, under 1 the conditions assumed .the reactive part of 21 does not vary as rapidly as the inverse firstpower of the frequency. If then 21 be made a negative capacitance there will be a frequency forwhich the real part of Z4/24 is greater than unity and therefore the curve includes the origin and the pedance of the line K the impedances 211 and 231 in Fig. 12 are also equal to K. The impedances 212 g and 232 are equal to K/tanh and the impedance Z2 is equal to K sinh'P where v P is the propagation constant of the line section between a and 12. See for example page 281 of K. S. Johnsons book. If we now attempt to make 24 equal to MK/ where M is a positive constant the expression for D reduces to the form P P M 1+tanh M 1+coth except for a factor made up of positive impedances which does not affect the question of whether the origin is included. Now it can be seen either by drawing the curves or by inspection in view of the known properties of tanh that the circuit is stable provided M is taken so large that for all frequencies, where A is the real part of P.

It thus appears that while negative capacitance cannot be employed for the purpose contemplated it is nevertheless possible to employ negative impedances when they are suitably chosen. It may be observed that when the midsection is short so that A approaches zero the permissible value of M approaches 1, but when the section is long so that A is great (and tanh A is near unity) the permissible value of M approaches 1/2. For intermediate lengths intermediate values are permissible. It should also be pointed out that M may be given values greater (by any amount) than these limiting values. Finally it should be remarked that impedance of forms other than 'MK/ are permissible, the criterion for any proposed impedance being whether or not the resulting curve of D encloses the origin.

It will be particularly understood that the various cases discussed hereinbefore are examples and in no way limit the scope of the invention, which is set forth in the appended claims.

What is claimed is:

1. The method of stabilizing a circuit including positive impedance elements and a negative impedance which consists in adjusting the constants of said circuit for a given frequency and measuring the positive impedance and the negative impedance, adjusting the circuit for another frequency and again measuring the positive and negative 'impedances, and continuing until the shape of theplot of such'measurements can be obtained for all frequencies from 'zeroto infinity,

plotting the locus of the real part of the ratio of the positive to the negative impedance against the imaginary part of such ratio at said frequen-' cies, and adjusting the constants of the :circuit until thesaid locus is a closed curve that excludes the point for which the real part of the ratio is unity and the imaginary part is zero.

2. The method of stabilizing a circuit which includes one or more positive impedances and one a or, morenegatiye impedances which comprises adjusting theconstants of said circuit for a given frequency and measuring the impedances to determine the quantity D as defined herein, adjusting the constants of said circuit for each of a number of other frequencies and measuring the positive and negative impedances necessary to determine the quantity Dr over a suiiicient range of frequencies so that the shape of the plot of suchmeasurements can be'obta'ined for all fre-' 'quencies from zeroto-infinity plottingthe locus of the real partagainst the ima'ginary part of the quantityrD at saidsfrequencies, and adjusting the constants of the circuit until the said locus is a closed curve which does'notinclude the origin. r 1

3. A series type negativeimpedance circuit hav-- ing a pair of terminals, a positive impedance cona negative impedanceof the series type, and the shunt arms being positiveimpedancesi 4. A shunt typefnegative impedance circuit having a pair of terminalsya'. positiveimpedance connected to the end of said. circuit opposite said terminals, a network. of. Ilconfiguration connected 7 in the'circuit between said impedance and'said:

terminals, the series arm'of said network being a a negative impedance of: the shunt type, and the shunt'arms being positive impedances. r r a HARRY NYQUI ST. 

